Similarity coefficient methods applied to the cell formation problem: a comparative investigation

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1 Computers & Industrial Engineering 48 (2005) Similarity coefficient methods applied to the cell formation problem: a comparative investigation Yong Yin a, *, Kazuhiko Yasuda b a Industrial and Management Science, Department of Public Policy and Social Studies, Yamagata University, , Kojirakawa-cho, Yamagata-shi , Japan b Graduate School of Economics and Management, Tohoku University, Kawauchi, Aoba-ku, Sendai , Japan Received 1 April 2001; revised 1 January 2003; accepted 1 January 2003 Abstract Although many similarity coefficients have been proposed, very few comparative studies have been done to evaluate the performance of various similarity coefficients. In this paper, we compare the performance of 20 wellknown similarity coefficients. Two hundred and fourteen numerical cell formation problems, which are selected from the literature or generated deliberately, are used for the comparative study. Nine performance measures are used for evaluating the goodness of cell formation solutions. Two characteristics, discriminability and stability of the similarity coefficients are tested under different data conditions. From the results, three similarity coefficients are found to be more discriminable. Jaccard is found to be the most stable similarity coefficient. Four similarity coefficients are not recommendable due to their poor performances. q 2005 Elsevier Ltd. All rights reserved. Keywords: Cellular manufacturing; Cell formation; Similarity coefficient; Comparative study 1. Introduction The design of cellular manufacturing systems is a complex, time-consuming work. Numerous methods have been proposed to solve the cell formation (CF) problem. Among these methods, those based on similarity coefficients are more basic and more flexible for dealing with the CF problem (Seifoddini, 1989a; Yin & Yasuda, 2005). * Corresponding author. Tel./fax: C address: yin@human.kj.yamagata-u.ac.jp (Y. Yin) /$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi: /j.cie

2 472 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Although numerous CF methods have been proposed, few comparative studies have been done to evaluate the robustness of various methods. Part reason is that different CF methods include different production factors, such as machine requirement, setup times, utilization, workload, setup cost, capacity, part alternative routings, and operation sequences. Selim, Askin, and Vakharia (1998) emphasized the necessity to evaluate and compare different CF methods based on the applicability, availability, and practicability. Previous comparative studies include Chu and Tsai (1990), Miltenburg and Zhang (1991), Mosier (1989), Seifoddini and Hsu (1994), Shafer and Meredith (1990), Shafer and Rogers (1993), and Vakharia and Wemmerlöv (1995). Among the above seven comparative studies, Chu and Tsai (1990) examined three array-based clustering algorithms: rank order clustering (ROC) (King, 1980), direct clustering analysis (DCA) (Chan & Milner, 1982), and bond energy analysis (BEA) (McCormick, Schweitzer, & White, 1972); Shafer and Meredith (1990) investigated six cell formation procedures: ROC, DCA, cluster identification algorithm (CIA) (Kusiak & Chow, 1987), single linkage clustering (SLC), average linkage clustering (ALC), and an operation sequences based similarity coefficient (Vakharia & Wemmerlöv, 1990); Miltenburg and Zhang (1991) compared nine cell formation procedures. Some of the compared procedures are combinations of two different algorithms A1/A2. A1/A2 denotes using A1 (algorithm 1) to group machines and using A2 (algorithm 2) to group parts. The nine procedures include: ROC, SLC/ ROC, SLC/SLC, ALC/ROC, ALC/ALC, modified ROC (MODROC) (Chandrasekharan & Rajagopalan, 1986b), ideal seed non-hierarchical clustering (ISNC) (Chandrasekharan & Rajagopalan, 1986a), SLC/ ISNC, and BEA. The other four comparative studies evaluated several similarity coefficients. We will discuss them in Section Comparative study 2.1. The objective of this study Although a large number of similarity coefficients exist in the literature, only a handful has been used for solving CF problems. Among various similarity coefficients, Jaccard similarity coefficient (Jaccard, 1908) was the most used similarity coefficient in the literature (Yin & Yasuda, 2005). However, contradictory viewpoints among researchers have been found in the previous studies: some researchers advocated the dominant power of Jaccard similarity coefficient; whereas some other researchers emphasized the drawbacks of Jaccard similarity coefficient and recommended other similarity coefficients; moreover, several researchers believed that there is no difference between Jaccard and other similarity coefficients, they considered that none of the similarity coefficients seems to perform always well under various cell formation situations. Therefore, a comparative research is crucially necessary to evaluate various similarity coefficients. Based on the comparative study, even if we cannot find a dominant similarity coefficient for all cell formation situations, at least we need to know which similarity coefficient is more efficient and more appropriate for some specific cell formation situations. In this paper, we investigate the performance of 20 well-known similarity coefficients. A large number of numerical data sets, which are taken from the open literature or generated specifically, are tested on nine performance measures.

3 2.2. Previous comparative studies Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Four studies that have focused on comparing various similarity coefficients and related cell formation procedures have been published in the literature. Mosier (1989) applied a mixture model experimental approach to compare seven similarity coefficients and four clustering algorithms. Four performance measures were used to judge the goodness of solutions: simple matching measure, generalized matching measure, product moment measure and intercellular transfer measure. As pointed out by Shafer and Rogers (1993), the limitation of this study is that three of the four performance measures are for measuring how closely the solution generated by the cell formation procedures matched the original machine-part matrix. However, the original machinepart matrix is not necessarily the best or even a good configuration. Only the last performance measure, intercellular transfer measure is for considering specific objectives associated with the CF problem. Shafer and Rogers (1993) compared 16 similarity coefficients and four clustering algorithms. Four performance measures were used to evaluate the solutions. Eleven small, binary machine-part group technology data sets mostly from the literature were used for the purpose of comparison. However, small and/or well-structured data sets may not have sufficient discriminatory power to separate good from inferior techniques. Further, results based on a small number of data sets may have little general reliability due to clustering results strong dependency on the input data (Anderberg, 1973; Milligan & Cooper, 1987; Vakharia & Wemmerlöv, 1995). Seifoddini and Hsu (1994) introduced a new performance measure: grouping capability index (GCI). The measure is based on exceptional elements and has been widely used in the subsequent research. However, only three similarity coefficients have been tested in their study. Vakharia and Wemmerlöv (1995) studied the impact of dissimilarity measures and clustering techniques on the quality of solutions in the context of cell formation. Twenty-four binary data sets were solved to evaluate eight dissimilarity measures and seven clustering algorithms. Some important insights have been provided by this study, such as data set characteristics, stopping parameters for clustering, performance measures, and the interaction between dissimilarity coefficients and clustering procedures. Unfortunately, similarity coefficients have not been discussed in this research. 3. Experimental design 3.1. The similarity coefficients tested in this study Twenty well-known similarity coefficients (Table 1) are compared in this paper. Among these similarity coefficients, several of them have never been studied in previous comparative research Data sets It is desirable to judge the effectiveness of various similarity coefficients under varying data set conditions. The tested data sets are classified into two distinct groups: selected from the literature and generated deliberately. Previous comparative studies used either one or the other to evaluate the performance of various similarity coefficients. Unlike those studies, this paper uses both types of the data sets to evaluate 20 similarity coefficients.

4 474 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Table 1 Definitions and ranges of selected similarity coefficients Coefficient Range Definition, S ij 1. Jaccard 0 1 a/(acbcc) 2. Hamann K1 to1 [(acd)k(bcc)]/[(acd)c(bcc)] 3. Yule K1 to1 (adkbc)/(adcbc) 4. Simple matching 0 1 (acd)/(acbcccd) 5. Sorenson 0 1 2a/(2aCbCc) 6. Rogers and Tanimoto 0 1 (acd)/[ac2(bcc)cd] 7. Sokal and Sneath 0 1 2(aCd)/[2(aCd)CbCc] 8. Rusell and Rao 0 1 a/(acbcccd) 9. Baroni-Urbani and Buser 0 1 [ac(ad) 1/2 ]/[acbccc(ad) 1/2 ] 10. Phi K1 to1 (adkbc)/[(acb)(acc)(bcd)(ccd) 1/2 ] 11. Ochiai 0 1 a/[(acb)(acc) 1/2 ] 12. PSC 0 1 a 2 /[(bca)(cca)] 13. Dot-product 0 1 a/(bccc2a) 14. Kulczynski 0 1 1/2[a/(aCb)Ca/(aCc)] 15. Sokal and Sneath a/[ac2(bcc)] 16. Sokal and Sneath /4[a/(aCb)Ca/(aCc)Cd/(bCd)Cd/(cCd)] 17. Relative matching 0 1 [ac(ad) 1/2 ]/[acbcccdc(ad) 1/2 ] 18. Chandrasekharan and Rajagopalan (1986b) 0 1 a/min[(acb),(acc)] 19. MaxSC 0 1 Max [a/(acb),a/(acc)] 20. Baker and Maropoulos (1997) 0 1 a/max[(acb),(acc)] a, the number of parts visit both machines; b, the number of parts visit machine i but not j; c, the number of parts visit machine j but not i; d, the number of parts visit neither machine Data sets selected from the literature In the previous comparative studies, Shafer and Rogers (1993), and Vakharia and Wemmerlöv (1995) took 11 and 24 binary data sets from the literature, respectively. The advantage of the data sets from the literature is that they stand for a variety of CF situations. In this paper, 70 data sets are selected from the literature. Table 2 shows the details of the 70 data sets Data sets generated deliberately From the computational experience with a wide variety of CF data sets, one finds that it may not always be possible to obtain a good GT solution, if the original CF problem is not amenable to wellstructural data set (Chandrasekharan & Rajagopalan, 1989). Hence, it is important to evaluate the quality of solutions of various structural data sets. Using data sets that are generated deliberately is a shortcut to evaluate the GT solutions obtained by various similarity coefficients. The generation process of data sets is often controlled by using experimental factors. In this paper, we use two experimental factors to generate data sets Ratio of non-zero element in cells (REC). Density is one of the most used experimental factors (Miltenburg & Zhang, 1991). However, in our opinion, density is an inappropriate factor for being used to control the generation process of cell formation data sets. We use Fig. 1 to illustrate this problem. Cell formation data are usually presented in a machine-part incidence matrix such as the one given in Fig. 1a. The matrix contains 0s and 1s elements that indicate the machine requirements of parts (to show

5 Table 2 Data sets from literature Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Data set Size NC a 1. Singh and Rajamani (1996) 4! Singh and Rajamani (1996) 4! Singh and Rajamani (1996) 5! Waghodekar and Sahu (1984) 5! Waghodekar and Sahu (1984) 5! Chow and Hawaleshka (1992) 5! Chow and Hawaleshka (1993a) 5! Chow and Hawaleshka (1993b) 5! Seifoddini (1989b) 5! Seifoddini (1989b) 5! Singh and Rajamani (1996) 6! Chen et al. (1996) 7! Boctor (1991) 7! Islam and Sarker (2000) 8! Seifoddini and Wolfe (1986) 8! Chandrasekharan and Rajagopalan (1986a) 8!20 2, Chandrasekharan and Rajagopalan (1986b) 8!20 2, Faber and Carter (1986) 9! Seifoddini and Wolfe (1986) 9! Chen et al. (1996) 9! Hon and Chi (1994) 9! Selvam and Balasubramanian (1985) 10! Mosier and Taube (1985a) 10! Seifoddini and Wolfe (1986) 10! McAuley (1972) 12! Seifoddini (1989a) 11! Hon and Chi (1994) 11! De Witte (1980) 12!19 2, Irani and Khator (1986) 14! Askin and Subramanian (1987) 14! King (1980) (machines 6, 8 removed) 14!43 4, Chan and Milner (1982) 15! Faber and Carter (1986) 16!16 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, Sofianopoulou (1997) 16!30 2, King (1980) 16!43 4, Boe and Cheng (1991) (mach 1 removed) 19! Shafer and Rogers (1993) 20! Shafer and Rogers (1993) 20!20 4 (continued on next page)

6 476 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Table 2 (continued) Data set Size NC a 48. Shafer and Rogers (1993) 20! Mosier and Taube (1985b) 20!20 3, Boe and Cheng (1991) 20! Ng (1993) 20! Kumar, Kusiak, and Vannelli (1986) 23!20 2, McCormick et al. (1972) 24! Carrie (1973) 24! Chandrasekharan and Rajagopalan (1989) 24! Chandrasekharan and Rajagopalan (1989) 24! Chandrasekharan and Rajagopalan (1989) 24! Chandrasekharan and Rajagopalan (1989) 24! Chandrasekharan and Rajagopalan (1989) 24! Chandrasekharan and Rajagopalan (1989) 24! Chandrasekharan and Rajagopalan (1989) 24! McCormick et al. (1972) 27! Carrie (1973) 28!46 3, Lee, Luong, and Abhary (1997) 30! Kumar and Vannelli (1987) 30!41 2, 3, Balasubramanian and Panneerselvam (1993) 36! King and Nakornchai (1982) 36!90 4, McCormick et al. (1972) 37!53 4, 5, Chandrasekharan and Rajagopalan (1987) 40! Seifoddini and Tjahjana (1999) 50!22 14 a Number of cells. the matrix clearly, 0s are not usually shown). Rows represent machines and columns represent parts. A 1 in the ith row and jth column represents that the jth part needs an operation on the ith machine; similarly, a 0 in the ith row and jth column represents that the ith machine is not needed to process the jth part. For Fig. 1a, we assume that two machine-cells exist. The first cell is constructed by machines 2, 4, 1 and parts 1, 3, 7, 6, 10; The second cell is constructed by machines 3, 5 and parts 2, 4, 8, 9, 5, 11. Without loss of generality, we use Fig. 1b to represent Fig. 1a. The two cells in Fig. 1a are now shown as capital letter A, we call A as the inside cell region. Similarly, we call B as the outside cell region. There are three densities that are called Problem Density (PD), non-zero elements Inside cells Density (ID) and non-zero elements Outside cells Density (OD). The calculations of these densities are as follows: total number of non-zero elements in regions A CB PD Z total number of elements in regions A CB (1) total number of non-zero elements in regions A ID Z total number of elements in regions A (2)

7 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Fig. 1. Illustration of three densities used by previous studies. total number of non-zero elements in regions B OD Z total number of elements in regions B (3) In the design of cellular manufacturing systems, what we are interested in is to find out appropriate machine-part cells the region A. In practice, region B is only a virtual region that does not exist in the real job shops. For example, if Fig. 1a is applied to a real-life job shop, Fig. 1c is a possible layout. There is no region B exists in the real-life job shop. Therefore, we conclude that region B based densities are meaningless. Since PD and OD are based on B, this drawback weakens the quality of generated data sets in the previous comparative studies. To overcome the above shortcoming, we introduce a ratio to replace the density used by previous researchers. The ratio is called as Ratio of non-zero Element in Cells (REC) and is defined as follows: total number of non-zero elements REC Z total number of elements in region A (4) The definition is intuitive. REC can also be used to estimate the productive capacity of machines. If REC is bigger than 1, current machine capacity cannot respond to the production requirements of parts. Thus, additional machines need to be considered. Therefore, REC can be used as a sensor to assess the capacity of machines Radio of exceptions (RE). The second experimental factor is Radio of Exceptions (RE). An exception is defined as a 1 in the region B (an operation outside the cell). We define RE as follows: total number of non-zero elements in region B RE Z total number of non-zero elements (5)

8 478 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Table 3 Test levels of REC and RE Level REC RE RE is used to judge the goodness of machine-part cells and distinguish well-structured problems from ill-structured problems. In this paper, three levels of REC, from sparse cells (0.70) to dense cells (0.90), and eight levels of RE, from well-structured cells (0.05) to ill-structured cells (0.40), are examined. 24 (3!8) combinations exist for all levels of the two experimental factors. For each combination, five 30!60-sized (30 machines by 60 parts) problems are generated. The generation process of the five problems is similar by using the random number. Therefore, a total of 120 test problems for all 24 combinations are generated, each problem is made up of six equally sized cells. The levels of REC and RE are shown in Table Clustering procedure The most well-known clustering procedures that have been applied to cell formation are single linkage clustering (SLC) algorithm, complete linkage clustering (CLC) algorithm and average linkage clustering (ALC) algorithm. These three procedures have been investigated by lots of studies. A summary of the past comparative results is shown in Table 4. Due to that ALC has the advantage of showing the greatest robustness regardless of similarity coefficients, in this paper, we select ALC as the clustering algorithm to evaluate the 20 similarity coefficients (Table 1). The ALC algorithm usually works as follows. Steps (1) Compute similarity coefficients for all machine pairs and store the values in a similarity matrix. (2) Join the two most similar objects (two machines, a machine and a machine group or two machine groups) to form a new machine group. Table 4 Comparative results of SLC, ALC and CLC Procedure Advantage Drawback SLC Simplicity; Minimal computational requirement; Tends to minimize the degree of adjusted machine duplication (Vakharia & Wemmerlöv, 1995) Largest tendency to chain; Leads to the lowest densities and the highest degree of single part cells (Gupta, 1991; Seifoddini, 1989a; Vakharia & Wemmerlöv, 1995) CLC Simplicity; Minimal computational requirement Performed as the worst procedure (Vakharia & ALC (does the reverse of SLC) Performed as the best procedure; Produces the lowest degree of chaining; Leads to the highest cell densities; Indifferent to choice of similarity coefficients; Few single part cells (Seifoddini, 1989a; Tarsuslugil & Bloor, 1979; Vakharia & Wemmerlöv, 1995; Yasuda & Yin, 2001) Wemmerlöv, 1995; Yasuda & Yin, 2001) Requires the highest degree of machine duplication; Requires more computation (Vakharia & Wemmerlöv, 1995)

9 (3) Evaluate the similarity coefficient between the new machine group and other remaining machine groups (machines) as follows: P P i2t j2v S ij S tv Z (6) N t N v where i is the machine in the machine group t; j is the machine in the machine group v. And N t is the number of machines in group t; N v is the number of machines in group v. (4) When all machines are grouped into a single machine group, or predetermined number of machine groups has been obtained, go to step 5; otherwise, go back to step 2. (5) Assign each part to the cell, in which the total number of exceptions is minimum Performance measures Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) A number of quantitative performance measures have been developed to evaluate the final cell formation solutions. Sarker and Mondal (1999) reviewed and compared various performance measures. Nine performance measures are used in this study to judge final solutions. These measures provide different viewpoints by judging solutions from different aspects Number of exceptional elements (EE) Exceptional elements are the source of inter-cell movements of parts. One objective of cell formation is to reduce the total cost of material handling. Therefore, EE is the most simple and intuitive measure for evaluating the cell formation solution Grouping efficiency Grouping efficiency is one of the first measures developed by Chandrasekharan and Rajagopalan (1986a, 1986b). Grouping efficiency is defined as a weighted average of two efficiencies h 1 and h 2 h Z wh 1 Cð1 KwÞh 2 (7) where h 1 Z o Ke o Ke Cv MP Ko Kv h 2 Z MP Ko Kv Ce M number of machines P number of parts o number of operations (1s) in the machine-part matrix {a ik } e number of exceptional elements in the solution v number of voids in the solution A value of 0.5 is recommended for w. h 1 is defined as the ratio of the number of 1s in the region A(Fig. 1b) to the total number of elements in the region A (both 0s and 1s). Similarly, h 2 is the ratio

10 480 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) of the number of 0s in the region B to the total number of elements in the region B (both 0s and 1s). The weighting factor allows the designer to alter the emphasis between utilization and inter-cell movement. The efficiency ranges from 0 to 1. Group efficiency that has been reported has a lower discriminating power (Chandrasekharan & Rajagopalan, 1987). Even an extremely bad solution with large number of exceptional elements has an efficiency value as high as Group efficacy To overcome the problem of group efficiency, Kumar and Chandrasekharan (1990) introduced a new measure, group efficacy t Z ð1 K4Þ=ð1 CfÞ where 4 is the ratio of the number of exceptional elements to the total number of elements; f is the ratio of the number of 0s in the region A to the total number of elements Machine utilization index (grouping measure, GM) Proposed by Miltenburg and Zhang (1991), which is used to measure machine utilization in a cell. The index is defined as follows h g Z h u Kh m (9) where h u Zd/(dCv) and h m Z1K(d/o). d is the number of 1s in the region A, h u is the measure of utilization of machines in a cell and h m is the measure of inter-cell movements of parts. h g ranges from K1 to1,h u and h m range from 0 to 1. A bigger value of machine utilization index h g is desired Clustering measure (CM) This measure tests how closely the 1s gather around the diagonal of the solution matrix, the definition of the measure is as follows (Singh & Rajamani, 1996) np MiZ1 P PkZ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio d 2 h ða ikþ Cd 2 vða ik Þ h c Z P MiZ1 P PkZ1 (10) a ik where d h (a ik ) and d v (a ik ) are horizontal and vertical distances between a non-zero entry a ik and the diagonal, respectively d h Z i K kðm K1Þ ðp KMÞ K ðp K1Þ ðp K1Þ (8) (11) iðp K1Þ d v Z k K ðm K1Þ K ðp KMÞ ðm K1Þ (12) Grouping index (GI) Nair and Narendran (1996) indicated that a good performance measure should be defined with reference to the block diagonal space. And the definition should ensure equal weightage to voids (0s in the region A) and exceptional elements. They introduced a measure, incorporating the block diagonal

11 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) space, weighting factor and correction factor g Z 1 K qvcð1kqþðekaþ B 1 C qvcð1kqþðekaþ (13) B where B is the block diagonal space and q is a weighting factor ranges between 0 and 1. AZ0 for e%b and AZeKB for eob. For convenience, Eq. (13) could be written as follows g Z 1 Ka (14) 1 Ca where qv Cð1 KqÞðe KAÞ a Z (15) B Both a and g range from 0 to Bond energy measure (BEM) McCormick et al. (1972) used the BEM to convert a binary matrix into a block diagonal form. This measure is defined as follows: P MiZ1 P PK1 kz1 a ik a iðkc1þ C P MK1 P PkZ1 iz1 a ik a ðic1þk h BE Z P MiZ1 P PkZ1 (16) a ik Bond energy is used to measure the relative clumpiness of a clustered matrix. Therefore, the more close the 1s are, the larger the bond energy measure will be Grouping capability index (GCI) Hsu (1990) showed that neither group efficiency nor group efficacy is consistent in predicting the performance of a cellular manufacturing system based on the structure of the corresponding machinepart matrix (Seifoddini & Djassemi, 1996). Hsu (1990) considered the GCI as follows: GCI Z 1 K e o (17) Unlike group efficiency and group efficacy, GCI excludes zero entries from the calculation of grouping efficacy Alternative routeing grouping efficiency (ARG efficiency) ARG was proposed by Sarker and Li (1998). ARG evaluates the grouping effect in the presence of alternative routings of parts. The efficiency is defined as follows 1 K e o 1 K v h ARG Z 0 z 0 Z o0 Ke z 0 Kv 1 C e o 1 C v o 0 Ce z 0 (18) Cv 0 z 0 where o 0 is the total number of 1s in the original machine-part incidence matrix with multiple process routings, z 0 is the total number of 0s in the original machine-part incidence matrix with multiple process

12 482 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) routings. ARG efficiency can also be used to evaluate CF problems that have no multiple process routings of parts. The efficiency ranges from 0 to 1 and is independent of the size of the problem. 4. Comparison and results Two key characteristics of similarity coefficients are tested in this study, discriminability and stability. In this study, we compare the similarity coefficients by using following steps. Comparative steps 1. Computation At first, solve each problem in the data sets by using 20 similarity coefficients; compute performance values by nine performance measures. Thus, we obtain at least a total of d!20!9 solutions. d is the number of the problems (some data sets from literature are multi-problems due to the different number of cells, see the item NC of Table 2) Average performance values matrix: create a matrix whose rows are problems and columns are nine performance measures. An element in row i and column j indicates, for problem i and performance measure j, the average performance value produced by 20 similarity coefficients. 2. Based on the results of step 1, construct two matrices whose rows are 20 similarity coefficients and columns are nine performance measures, an entry SM ij in the matrices indicates: 2.1. Discriminability matrix: the number of problems in which the similarity coefficient i gives the best performance value for measure j Stability matrix: the number of problems in which the similarity coefficient i gives the performance value of measure j with at least average value (better or equal than the value in the matrix of step 1.2). 3. For each performance measure, find the top five values in the above two matrices. The similarity coefficients corresponding to these values are considered to be the most discriminable/stable similarity coefficients for this performance measure. 4. Based on the results of step 3, for each similarity coefficient, find the number of times that it has been selected as the most discriminable/stable coefficient for the total nine performance measures. We use small examples here to show the comparative steps. Step 1.1: a total of 214 problems were solved. One hundred and twenty problems were deliberately generated; 94 problems were from literature, see Table 2 (some data sets were multi-problems due to the different number of cells). A total of 38,520 (214!20!9) performance values were obtained by using 20 similarity coefficients and nine performance measures. For example, by using Jaccard similarity coefficient, the nine performance values of the problem McCormick et al. (no. 62 in Table 2) are as follows (Table 5). Table 5 The performance values of McCormick et al. by using Jaccard similarity coefficient EE Grouping efficiency Group efficacy GM CM GI BEM GCI ARG Jaccard

13 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Table 6 The average performance values of 20 similarity coefficients, for the problem McCormick et al. EE Grouping efficiency Group efficacy GM CM GI BEM GCI ARG Average values Step 1.2: The average performance values matrix contained 214 problems (rows) and nine performance measures (columns). An example of row (problem McCormick et al.) is as follows (Table 6). We use Jaccard similarity coefficient and the 94 problems from literature to explain steps 2 4. Step 2.1 (discriminability matrix): among the 94 problems and for each performance measure, the numbers of problems in which Jaccard s similarity coefficient gave the best values are shown in Table 7. For example, the 60 in the column EE means that comparing with other 19 similarity coefficients, Jaccard produced minimum exceptional elements to 60 problems. Step 2.2 (stability matrix): among the 94 problems and for each performance measure, the number of problems in which Jaccard s similarity coefficient gave the value with at least average value (matrix of step 1.2) are shown in Table 8. For example, the meaning of 85 in the column EE is as follows: comparing with the average exceptional elements of 94 problems in the matrix of step 1.2, the number of problems in which Jaccard produced fewer exceptional elements are 85. Step 3: For example, for the exceptional elements, the similarity coefficients that corresponded to the top five values in the discriminability matrix are Jaccard, Sorenson, Rusell and Rao, Dot-product, Sokal and Sneath 2, Relative matching, and Baker and Maropoulos. These similarity coefficients are considered as the most discriminable coefficients for the performance measure-exceptional elements. The same procedures are conducted to the other performance measures and stability matrix. Step 4: Using the results of step 3, Jaccard has been selected 5/6 times as the most discriminable/stable similarity coefficient. That means, among nine performance measures, Jaccard is the most discriminable/stable similarity coefficient for 5/6 performance measures. The result is shown in the column literature of Table 9. The results are shown in Table 9 and Figs. 2 4 (in the figures, the horizontal axes are 20 similarity coefficients and the vertical axes are nine performance measures). The tables and figures show the number of performance measures for which these 20 similarity coefficients have been regarded Table 7 The number of problems in which Jaccard s similarity coefficient gave the best performance values EE Grouping efficiency Group efficacy GM CM GI BEM GCI ARG Jaccard Table 8 The number of problems in which Jaccard s similarity coefficient gave the best performance values EE Grouping efficiency Group efficacy GM CM GI BEM GCI ARG Jaccard

14 Table 9 Comparative results under various conditions No. Similarity coefficient Literature All random REC RE D S D S D S D S D S D S D S D S 1 Jaccard Hamann Yule Simple matching 5 Sorenson Rogers and Tanimoto 7 Sokal and Sneath 8 Rusell and Rao 9 Baoroni Urban and Buser 10 Phi Ochiai PSC Dot-product Kulczynski Sokal and Sneath 2 16 Sokal and Sneath 4 17 Relative matching 18 Chandraseharan and Rajagopalan 19 MaxSC Baker and Maropoulos D, discriminability; S, stability. 484 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005)

15 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Fig. 2. Performance for all tested problems. as the most discriminable/stable coefficients. The columns of the table represent different conditions of data sets. The column literature includes all 94 problems from literature; the column all random includes all 120 deliberately generated problems. The deliberately generated problems are further investigated by using different levels of REC and RE. Fig. 3. Performance under different REC.

16 486 Y. Yin, K. Yasuda / Computers & Industrial Engineering 48 (2005) Fig. 4. Performance under different RE. Literature and All random columns in Table 9 (also Fig. 2) give the performance results of all 214 tested problems. We can find that Jaccard and Sorenson are two best coefficients. On the other hand, four similarity coefficients: Hamann, Simple matching, Rogers and Tanimoto, and Sokal and Sneath are inefficient in both discriminability and stability. REC columns in Table 9 (also Fig. 3) show the performance results under the condition of different REC ratios. We can find that almost all similarity coefficients perform well under a high REC ratio. However, four similarity coefficients: Hamann, Simple matching, Rogers and Tanimoto, and Sokal and Sneath, again produce bad results under the low REC ratio. RE columns in Table 9 (also Fig. 4) give the performance results under the condition of different RE ratios. All similarity coefficients perform best under a low RE ratio (data sets are wellstructured). Only a few of similarity coefficients perform well under a high RE ratio (data sets are illstructured), Sokal and Sneath 2 is very good for all RE ratios. Again, the four similarity coefficients: Hamann, Simple matching, Rogers and Tanimoto, and Sokal and Sneath, perform badly under high RE ratios. In summary, three similarity coefficients: Jaccard, Sorenson, and Sokal and Sneath 2 perform best among twenty tested similarity coefficients. Jaccard emerges from the 20 similarity coefficients for its stability. For all problems, from literature or deliberately generated; and for all levels of both REC and RE ratios, Jaccard similarity coefficient is constantly the most stable coefficient among all 20 similarity coefficients. Another finding in this study is four similarity coefficients: Hamann, Simple matching, Rogers and Tanimoto, and Sokal and Sneath are inefficient under all conditions. So, these similarity coefficients are not recommendable for cell formation applications.

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